Problem 43 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.2, review translating sentences to equations (Section 1.1). Translate each sentence to an equation.\([1.1]\) The sum of three consecutive numbers is 100.

Step-by-Step Solution

Answer

The sum of three consecutive numbers cannot be 100.

Step 1: Identify Consecutive Numbers

Consecutive numbers follow one another in order, so if the first number is x, the second number would be x+1, and the third number would be x+2.

Step 2: Set Up the Equation

According to the problem, the sum of these three consecutive numbers is 100. This means we need to add them together and set the sum equal to 100. So, the equation will be: \[ x + (x + 1) + (x + 2) = 100 \]

Step 3: Simplify the Equation

Combine the like terms in the equation: \[ x + x + 1 + x + 2 = 100 \] Simplify to get: \[ 3x + 3 = 100 \]

Step 4: Solve for x

To isolate x, first subtract 3 from both sides of the equation: \[ 3x + 3 - 3 = 100 - 3 \] Which simplifies to: \[ 3x = 97 \] Then, divide both sides by 3: \[ x = \frac{97}{3} \] Which results to: \[ x = 32.33 \]

Key Concepts

consecutive numbers

Consecutive numbers are numbers that follow each other in sequence. In simpler terms, they are numbers that come one after another without any numbers skipped in between. For example, the numbers 4, 5, and 6 are consecutive. When we talk about consecutive numbers in an algebraic context, we often use variables to represent them in equations.
For instance, if the first number is represented by \( x \), the next consecutive number will be \( x+1 \) and the next one would be \( x+2 \), and so on. This helps in simplifying the process of translating the problem statement into an equation.
In the exercise provided, the first number is \( x \), the second is \( x+1 \), and the third is \( x+2 \). By identifying these consecutive numbers, we can then set up our equation with these variables.

solving equations

Solving equations involves finding the value of the variable that makes the equation true. Here are the steps to solve the equation effectively:

• Set Up the Equation: Translate the word problem into an equation using identified variables. For example, we found that the sum of \( x, x+1, \text {and} x+2 \) is 100. This results in the equation \[ x + (x + 1) + (x + 2) = 100 \].
• Combine Like Terms: Simplify the equation by adding the coefficients of the same variables. In our case, combining like terms, \( x + x + x\), simplifies the equation to \[ 3x + 3 = 100 \].
• Isolate the Variable: Solve for the variable by performing operations such as addition, subtraction, multiplication, and division. First, subtract 3 from both sides: \[ 3x + 3 - 3 = 100 - 3 \] which simplifies to \[ 3x = 97 \]. Next, divide both sides by 3: \[ x = \frac{97}{3} \].
• Check the Solution: Verify the solution to ensure that it satisfies the original equation. Substituting \( x = 32.33 \) back into the equation ensures that the equation holds true.

By following these steps, you can solve any basic algebraic equation you encounter.

combining like terms

Combining like terms is a crucial step in simplifying algebraic expressions and equations. It involves combining terms that have the same variables raised to the same power. This makes the equation simpler and more manageable to solve.
When given an equation like \[ x + (x + 1) + (x + 2) = 100 \], we can combine the terms with the same variable. In this example, the terms \( x \), \( x \), and \( x \) are all like terms. Adding these together gives us \[ 3x \].
Next, we add the constant terms (numbers without variables). The constants in this equation are 1 and 2. Adding these gives us \[ 1 + 2 = 3 \]. Therefore, the simplified equation becomes \[ 3x + 3 = 100 \].
By following these steps, one can easily handle algebraic expressions and make them simpler to solve. This process significantly helps in isolating the variable and finding the solution efficiently.